To find the limit of the function [tex]\(\frac{x^2 \cdot x}{x^3 - x}\)[/tex] as [tex]\(x\)[/tex] approaches 6, follow these steps:
1. Simplify the Expression:
First, rewrite the given expression in a more simplified form:
[tex]\[
\frac{x^2 \cdot x}{x^3 - x} = \frac{x^3}{x^3 - x}
\][/tex]
2. Factor the Denominator:
Notice that the denominator can be factored using the difference of cubes:
[tex]\[
x^3 - x = x(x^2 - 1) = x(x - 1)(x + 1)
\][/tex]
3. Rewrite the Expression with Factored Denominator:
Substitute the factored form of the denominator back into the expression:
[tex]\[
\frac{x^3}{x(x - 1)(x + 1)}
\][/tex]
4. Cancel Common Factors:
Simplify the fraction by canceling the common factor [tex]\(x\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{x^3}{x(x - 1)(x + 1)} = \frac{x^2}{(x - 1)(x + 1)}
\][/tex]
5. Substitute [tex]\(x = 6\)[/tex]:
Now, we can substitute [tex]\(x = 6\)[/tex] directly into the simplified expression:
[tex]\[
\frac{6^2}{(6 - 1)(6 + 1)} = \frac{36}{5 \cdot 7} = \frac{36}{35}
\][/tex]
Therefore, the limit is:
[tex]\[
\operatorname{Lim}_{x \rightarrow 6} \frac{x^2 \cdot x}{x^3 - x} = \frac{36}{35}
\][/tex]
